Solid color for each face in cycles when using image textures?
Here is one way to do this :


duplicate the final mesh (modifiers applied )
UV unwrap the new object using this method :


select all face
hit Ctrl+E and select Edges Split
resize all faces to Zero with Individual origins as a pivot 
U and select Project from View Bounds






copy this map to the original object, using CtrlL>Transfer UV map


After that all points on the same face will have the same UV coordinates meaning they will have the same color in render.
You can adjust the UVmap for more precise output.



For the workflow of setting up a good mesh  :


Add the Image as a background in the view
Add a plane (or simply import the same image as plane to skip setting up materials)
Delete the face (in Edit mode Select All and X and select Only Faces)
Start doing a loose tracing of the image by adding vertices in Edit mode (Ctrl+LMB, don't worry about crossing edges!)





Delete all edges (Select All and X and select Only Edges&amp;Faces)
For this step you need the Delaunay/voronoi Addon it will create a triangulated mesh from the points 





The rest is described in the first part above 


And here is a sweet penguin render :